Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-2x+2y &= 4 \\ 8x+2y &= 6\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}2x-2y &= -4\\ 8x+2y &= 6\end{align*}$ Add the top and bottom equations. $10x = 2$ Divide both sides by $10$ and reduce as necessary. $x = \dfrac{1}{5}$ Substitute $\dfrac{1}{5}$ for $x$ in the top equation. $-2( \dfrac{1}{5})+2y = 4$ $-\dfrac{2}{5}+2y = 4$ $2y = \dfrac{22}{5}$ $y = \dfrac{11}{5}$ The solution is $\enspace x = \dfrac{1}{5}, \enspace y = \dfrac{11}{5}$.